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Spectral sequences in string topology

Lennart Meier

Algebraic & Geometric Topology 11 (2011) 2829–2860

In this paper, we investigate the behavior of the Serre spectral sequence with respect to the algebraic structures of string topology in generalized homology theories, specifically with the Chas–Sullivan product and the corresponding coproduct and module structures. We prove compatibility for two kinds of fiber bundles: the fiber bundle ΩnM LnM M for an h–oriented manifold M and the looped fiber bundle LnF LnE LnB of a fiber bundle F E B of h–oriented manifolds. Our method lies in the construction of Gysin morphisms of spectral sequences. We apply these results to study the ordinary homology of the free loop spaces of sphere bundles and some generalized homologies of the free loop spaces of spheres and projective spaces. For the latter purpose, we construct explicit manifold generators for the homology of these spaces.

string topology, free loop space, Serre spectral sequence, Gysin morphism
Mathematical Subject Classification 2000
Primary: 55P35, 55T10
Secondary: 57R19
Received: 8 June 2010
Revised: 12 September 2011
Accepted: 14 September 2011
Published: 14 October 2011
Lennart Meier
Mathematisches Institut
Universität Bonn
Endenicher Allee 60
D-53115 Bonn